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Let $\mathcal{E}=⟨|\mathbf{u}{|}^2⟩/2$ be the turbulent kinetic energy per unit mass. In addition to $L>0$ and high Reynolds number, use the usual self-preserving large-eddy decay assumption: the normalized large-scale correlation has a fixed shape, sowhere $C_L>0$ is constant. For freely decaying high-Reynolds-number turbulence, the cascade viscous dissipation law is $ϵ=C_{ϵ}{\mathcal{E}}^{3/2}/\ell$. The kinetic-energy balance therefore givesPut $A=C_{ϵ}{C}_L^{1/3}{L}^{-1/3}$. Integrating,The constant $t_{\ast }$ is a virtual time origin. This is the Saffman decay law. The energy decreases while the integral scale of turbulence grows so that the conserved large-volume momentum variance remains constant.

The exponent does not follow from the invariant alone: it also uses unforced decay, one evolving large-eddy length and an asymptotically constant cascade viscous dissipation coefficient. A changing normalized spectral shape or a confining boundary can invalidate that closure. Moreover $\mathrm{Re}\sim {\mathcal{E}}^{1/2}\ell /\nu \propto (t-t_{\ast }{)}^{-1/5}$, so at fixed kinematic viscosity the high-Reynolds-number decay regime eventually ends; the $-6/5$ law is not a claim about the ultimate viscous limit.